Single-Valued $q$-Differentials on Variable Finite Riemann Surface

In present article are construction general theory of single-valued functions and $q$-differentials on variable finite Riemann surfaces begun. We constructed all form elementary $q$-differentials. We found dimensions and constructed explicit basis in important two factor spaces. In particular, we found dimension and constructed basis in first holomorphic cohomology group de Rham for 1-differentials.